Note on vanishing power sums of roots of unity

TL;DR

This paper characterizes all positive integers n for which there exist mth roots of unity summing to zero when raised to the ℓth power, extending previous results and exploring specific cases for n between 2 and m.

Contribution

It provides a complete classification of such n for given m and ℓ, extending earlier work on vanishing sums of roots of unity.

Findings

Complete list of n with roots of unity sum zero for given m and ℓ

Extension of Lam and Leung's results

Identification of roots of unity sums for 2 ≤ n ≤ m

Abstract

For a given positive integers $m$ and $\ell$, we give a complete list of positive integers $n$ for which their exist $m$th roots of unity $x_1,\dots,x_n \in \mathbb{C}$ such that $x_1^{\ell} + \cdots + x_n^{\ell}=0$. This extends the earlier result of Lam and Leung on vanishing sums of roots of unity. Furthermore, we prove that for which integers $n$ with $2 \leq n \leq m$, there are distinct $m$th roots of unity $x_1,\dots,x_n \in \mathbb{C}$ such that $x_1^{\ell} + \cdots + x_n^{\ell}=0$.